Science:Math Exam Resources/Courses/MATH257/December 2011/Question 03 (b)
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Question 03 (b) 

(b) According to the Fourier Convergence Theorem, for which values of in the interval should each of these series converge to ? Verify your conclusions for the Fourier sine series at and . 
Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you? 
If you are stuck, check the hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint. 
Hint 1 

What does the Fourier convergence theorem say? If you are not sure, you can start by look it up. It can be found in your textbook (Boyce & DiPrima). It probably is in your class notes as well. 
Hint 2 

It is a good idea to sketch the odd and even extensions of the function . Do you know why? 
Hint 3 

The reason we want to sketch both the odd and even extensions of is because the Fourier sine series comes from the odd extension of , while the Fourier cosine series comes from the even extension of . 
Checking a solution serves two purposes: helping you if, after having used all the hints, you still are stuck on the problem; or if you have solved the problem and would like to check your work.

Solution  

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Please rate my easiness! It's quick and helps everyone guide their studies. In short, the Fourier convergence theorem states the following.
In the above, denotes the right hand limit of at , while denotes the left hand limit of at . In other words, for a real number ,
The graph of is obtained by first reflecting over the yaxis and hence obtaining a function over the interval . We then periodic extend this function to get which will have a period of .
The graph of is obtained by first rotating about the origin by 180 degree and hence obtaining a function over the interval . We then periodic extend this function to get which will have a period of . From the sketch of is discontinuous at points with k an integer. Hence, by the Fourier convergence theorem, the Fourier sine series converges to at every point in the interval , but at , converges to
Since agrees with in the interval ,
and evaluating at , we get as expected. 